3.23 \(\int \frac{x^5}{(a+b \sec (c+d x^2))^2} \, dx\)

Optimal. Leaf size=1092 \[ \text{result too large to display} \]

[Out]

((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - I*Sqrt[a^2 -
b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b
^2)*d^2) - ((I/2)*b^3*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) +
(I*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^4*Log[
1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^4*Log[1 + (a*E^(I*(c +
d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - I*
Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2]))
])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 +
b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*
d^2) + (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (2
*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (I*b^3*PolyLo
g[3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, -((a
*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, -((a*E^(I*(c + d*
x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b
+ Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (b^2*x^4*Sin[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c
+ d*x^2]))

________________________________________________________________________________________

Rubi [A]  time = 2.2877, antiderivative size = 1092, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4204, 4191, 3324, 3321, 2264, 2190, 2531, 2282, 6589, 4522, 2279, 2391} \[ \frac{x^6}{6 a^2}+\frac{i b \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{i b^3 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b-\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}-\frac{i b \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{i b^3 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b+\sqrt{b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac{b^2 \sin \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (d x^2+c\right )\right )}-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b-i \sqrt{a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 \log \left (\frac{e^{i \left (d x^2+c\right )} a}{b+i \sqrt{a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{2 b \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac{i b^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+c\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{2 i b \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}-\frac{i b^3 \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+c\right )}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac{2 i b \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} d^3}+\frac{i b^3 \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+c\right )}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*Sec[c + d*x^2])^2,x]

[Out]

((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - I*Sqrt[a^2 -
b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b
^2)*d^2) - ((I/2)*b^3*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) +
(I*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^4*Log[
1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^4*Log[1 + (a*E^(I*(c +
d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - I*
Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2]))
])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 +
b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*
d^2) + (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (2
*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (I*b^3*PolyLo
g[3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, -((a
*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, -((a*E^(I*(c + d*
x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b
+ Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (b^2*x^4*Sin[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c
+ d*x^2]))

Rule 4204

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3321

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c
+ d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(
2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4522

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^2, 2] + I
*b*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b \sec (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \cos (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac{x^6}{6 a^2}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{(b+a \cos (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac{x^6}{6 a^2}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x^2}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x \sin (c+d x)}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}-\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right )}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt{-a^2+b^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt{-a^2+b^2}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b-\sqrt{a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x}{i b+\sqrt{a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}+\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^2}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{i a e^{i (c+d x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+\frac{i a e^{i (c+d x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{(2 i b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{(2 i b) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i a x}{i b-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i a x}{i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{(2 b) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}+\frac{(2 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{(2 i b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{b^3 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{2 i b \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}-\frac{2 i b \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=-\frac{i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac{x^6}{6 a^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac{b^2 x^2 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}+\frac{i b^3 x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac{i b x^4 \log \left (1+\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{i b^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}+\frac{b^3 x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac{2 b x^2 \text{Li}_2\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^2}-\frac{i b^3 \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac{2 i b \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{i b^3 \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac{2 i b \text{Li}_3\left (-\frac{a e^{i \left (c+d x^2\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} d^3}+\frac{b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )}\\ \end{align*}

Mathematica [A]  time = 6.16939, size = 895, normalized size = 0.82 \[ \frac{\left (b+a \cos \left (d x^2+c\right )\right ) \sec ^2\left (d x^2+c\right ) \left (\left (b+a \cos \left (d x^2+c\right )\right ) x^6+\frac{3 b^2 \left (a \sin \left (d x^2\right )-b \sin (c)\right ) x^4}{(a-b) (a+b) d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right )}-\frac{3 b \left (b+a \cos \left (d x^2+c\right )\right ) \left (2 \left (1+e^{2 i c}\right ) \left (-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2+i b \sqrt{\left (b^2-a^2\right ) e^{2 i c}}\right ) \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+2 c\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )+2 \left (1+e^{2 i c}\right ) \left (2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2+i b \sqrt{\left (b^2-a^2\right ) e^{2 i c}}\right ) \text{PolyLog}\left (2,-\frac{a e^{i \left (d x^2+2 c\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )+i \left (d \left (2 b d e^{2 i c} \sqrt{\left (b^2-a^2\right ) e^{2 i c}} x^2+\left (1+e^{2 i c}\right ) \left (-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2+2 i b \sqrt{\left (b^2-a^2\right ) e^{2 i c}}\right ) \log \left (\frac{e^{i \left (d x^2+2 c\right )} a}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}+1\right )+\left (1+e^{2 i c}\right ) \left (2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2+2 i b \sqrt{\left (b^2-a^2\right ) e^{2 i c}}\right ) \log \left (\frac{e^{i \left (d x^2+2 c\right )} a}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}+1\right )\right ) x^2-2 \left (2 a^2-b^2\right ) e^{i c} \left (1+e^{2 i c}\right ) \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+2 c\right )}}{b e^{i c}-\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )+2 \left (2 a^2-b^2\right ) e^{i c} \left (1+e^{2 i c}\right ) \text{PolyLog}\left (3,-\frac{a e^{i \left (d x^2+2 c\right )}}{e^{i c} b+\sqrt{\left (b^2-a^2\right ) e^{2 i c}}}\right )\right )\right )}{\left (a^2-b^2\right ) d^3 \sqrt{\left (b^2-a^2\right ) e^{2 i c}} \left (1+e^{2 i c}\right )}\right )}{6 a^2 \left (a+b \sec \left (d x^2+c\right )\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/(a + b*Sec[c + d*x^2])^2,x]

[Out]

((b + a*Cos[c + d*x^2])*Sec[c + d*x^2]^2*(x^6*(b + a*Cos[c + d*x^2]) - (3*b*(b + a*Cos[c + d*x^2])*(2*(1 + E^(
(2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*
(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(1 + E^((2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^
((2*I)*c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(
-a^2 + b^2)*E^((2*I)*c)]))] + I*(d*x^2*(2*b*d*E^((2*I)*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2 + (1 + E^((2*I)*c
))*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*Log[1 + (a*E^(I*(2*c + d
*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (1 + E^((2*I)*c))*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*
c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d*E^(I*c)*x^2)*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*
E^((2*I)*c)])]) - 2*(2*a^2 - b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) -
Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(2*a^2 - b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*(2*c + d*x^
2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])))/((a^2 - b^2)*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*(1 + E^
((2*I)*c))) + (3*b^2*x^4*(-(b*Sin[c]) + a*Sin[d*x^2]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Si
n[c/2]))))/(6*a^2*(a + b*Sec[c + d*x^2])^2)

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{ \left ( a+b\sec \left ( d{x}^{2}+c \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b*sec(d*x^2+c))^2,x)

[Out]

int(x^5/(a+b*sec(d*x^2+c))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/6*((a^4 - a^2*b^2)*d*x^6*cos(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6*cos(d*x^2 + c)^2 + (a^4 - a^2*b^2)*d
*x^6*sin(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6*sin(d*x^2 + c)^2 + 4*(a^3*b - a*b^3)*d*x^6*cos(d*x^2 + c)
+ 6*a*b^3*x^4*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6 + 2*(2*(a^3*b - a*b^3)*d*x^6*cos(d*x^2 + c) - 3*a*b^3*x^4
*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6)*cos(2*d*x^2 + 2*c) - 6*((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a
^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(2*d*
x^2 + 2*c)*sin(d*x^2 + c) + 4*(a^4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) +
(a^6 - a^4*b^2)*d + 2*(2*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c))*integrate
(2*(2*(2*a^2*b^2 - b^4)*d*x^5*cos(d*x^2 + c)^2 + 2*(2*a^2*b^2 - b^4)*d*x^5*sin(d*x^2 + c)^2 + (2*a^3*b - a*b^3
)*d*x^5*cos(d*x^2 + c) + 2*a*b^3*x^3*sin(d*x^2 + c) + ((2*a^3*b - a*b^3)*d*x^5*cos(d*x^2 + c) - 2*a*b^3*x^3*si
n(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (2*a*b^3*x^3*cos(d*x^2 + c) + (2*a^3*b - a*b^3)*d*x^5*sin(d*x^2 + c) + 2*a^
2*b^2*x^3)*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c
)^2 + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^5*b - a^3*b^3)*d*sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + 4*(a^
4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d + 2*(2*(a^5*b -
a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c)), x) + 2*(3*a*b^3*x^4*cos(d*x^2 + c) + 2*(a
^3*b - a*b^3)*d*x^6*sin(d*x^2 + c) + 3*a^2*b^2*x^4)*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^
2 + 4*(a^4*b^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^5*b - a^3*b^3)*d*
sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + 4*(a^4*b^2 - a^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2
+ c) + (a^6 - a^4*b^2)*d + 2*(2*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d)*cos(2*d*x^2 + 2*c))

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Fricas [C]  time = 3.77734, size = 6750, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="fricas")

[Out]

1/24*(4*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6*cos(d*x^2 + c) + 4*(a^4*b - 2*a^2*b^3 + b^5)*d^3*x^6 + 12*(a^3*b^2 -
a*b^4)*d^2*x^4*sin(d*x^2 + c) + 2*(-12*I*a^3*b^2 + 6*I*a*b^4 + (-12*I*a^4*b + 6*I*a^2*b^3)*cos(d*x^2 + c))*sq
rt(-(a^2 - b^2)/a^2)*polylog(3, -(b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 +
c))*sqrt(-(a^2 - b^2)/a^2))/a) + 2*(12*I*a^3*b^2 - 6*I*a*b^4 + (12*I*a^4*b - 6*I*a^2*b^3)*cos(d*x^2 + c))*sqrt
(-(a^2 - b^2)/a^2)*polylog(3, -(b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c)
)*sqrt(-(a^2 - b^2)/a^2))/a) + 2*(12*I*a^3*b^2 - 6*I*a*b^4 + (12*I*a^4*b - 6*I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-
(a^2 - b^2)/a^2)*polylog(3, -(b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*
sqrt(-(a^2 - b^2)/a^2))/a) + 2*(-12*I*a^3*b^2 + 6*I*a*b^4 + (-12*I*a^4*b + 6*I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-
(a^2 - b^2)/a^2)*polylog(3, -(b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*
sqrt(-(a^2 - b^2)/a^2))/a) + (-12*I*a^2*b^3 + 12*I*b^5 + (-12*I*a^3*b^2 + 12*I*a*b^4)*cos(d*x^2 + c) - 12*((2*
a^4*b - a^2*b^3)*d*x^2*cos(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*d*x^2)*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos
(d*x^2 + c) + 2*I*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a
+ 1) + (-12*I*a^2*b^3 + 12*I*b^5 + (-12*I*a^3*b^2 + 12*I*a*b^4)*cos(d*x^2 + c) + 12*((2*a^4*b - a^2*b^3)*d*x^
2*cos(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*d*x^2)*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(d*x^2 + c) + 2*I*b*s
in(d*x^2 + c) - 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) + (12*I*a^2*b^3
- 12*I*b^5 + (12*I*a^3*b^2 - 12*I*a*b^4)*cos(d*x^2 + c) - 12*((2*a^4*b - a^2*b^3)*d*x^2*cos(d*x^2 + c) + (2*a
^3*b^2 - a*b^4)*d*x^2)*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) + 2*(a*co
s(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) + (12*I*a^2*b^3 - 12*I*b^5 + (12*I*a^3
*b^2 - 12*I*a*b^4)*cos(d*x^2 + c) + 12*((2*a^4*b - a^2*b^3)*d*x^2*cos(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*d*x^2)*
sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) - I*a*sin(
d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) - 2*(6*(a^3*b^2 - a*b^4)*c*cos(d*x^2 + c) + 6*(a^2*b^3 - b^5)
*c - (3*I*(2*a^4*b - a^2*b^3)*c^2*cos(d*x^2 + c) + 3*I*(2*a^3*b^2 - a*b^4)*c^2)*sqrt(-(a^2 - b^2)/a^2))*log(2*
a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 2*(6*(a^3*b^2 - a*b^4)*c*cos(d*x
^2 + c) + 6*(a^2*b^3 - b^5)*c - (-3*I*(2*a^4*b - a^2*b^3)*c^2*cos(d*x^2 + c) - 3*I*(2*a^3*b^2 - a*b^4)*c^2)*sq
rt(-(a^2 - b^2)/a^2))*log(2*a*cos(d*x^2 + c) - 2*I*a*sin(d*x^2 + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 2*(6
*(a^3*b^2 - a*b^4)*c*cos(d*x^2 + c) + 6*(a^2*b^3 - b^5)*c - (3*I*(2*a^4*b - a^2*b^3)*c^2*cos(d*x^2 + c) + 3*I*
(2*a^3*b^2 - a*b^4)*c^2)*sqrt(-(a^2 - b^2)/a^2))*log(-2*a*cos(d*x^2 + c) + 2*I*a*sin(d*x^2 + c) + 2*a*sqrt(-(a
^2 - b^2)/a^2) - 2*b) - 2*(6*(a^3*b^2 - a*b^4)*c*cos(d*x^2 + c) + 6*(a^2*b^3 - b^5)*c - (-3*I*(2*a^4*b - a^2*b
^3)*c^2*cos(d*x^2 + c) - 3*I*(2*a^3*b^2 - a*b^4)*c^2)*sqrt(-(a^2 - b^2)/a^2))*log(-2*a*cos(d*x^2 + c) - 2*I*a*
sin(d*x^2 + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) + 2*(6*(a^2*b^3 - b^5)*d*x^2 + 6*(a^2*b^3 - b^5)*c + 6*((a^
3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*cos(d*x^2 + c) + (-3*I*(2*a^3*b^2 - a*b^4)*d^2*x^4 + 3*I*(2*a^3*b^
2 - a*b^4)*c^2 + (-3*I*(2*a^4*b - a^2*b^3)*d^2*x^4 + 3*I*(2*a^4*b - a^2*b^3)*c^2)*cos(d*x^2 + c))*sqrt(-(a^2 -
b^2)/a^2))*log(1/2*(2*b*cos(d*x^2 + c) + 2*I*b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqr
t(-(a^2 - b^2)/a^2) + 2*a)/a) + 2*(6*(a^2*b^3 - b^5)*d*x^2 + 6*(a^2*b^3 - b^5)*c + 6*((a^3*b^2 - a*b^4)*d*x^2
+ (a^3*b^2 - a*b^4)*c)*cos(d*x^2 + c) + (3*I*(2*a^3*b^2 - a*b^4)*d^2*x^4 - 3*I*(2*a^3*b^2 - a*b^4)*c^2 + (3*I*
(2*a^4*b - a^2*b^3)*d^2*x^4 - 3*I*(2*a^4*b - a^2*b^3)*c^2)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log(1/2*(2*
b*cos(d*x^2 + c) + 2*I*b*sin(d*x^2 + c) - 2*(a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2
*a)/a) + 2*(6*(a^2*b^3 - b^5)*d*x^2 + 6*(a^2*b^3 - b^5)*c + 6*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*
cos(d*x^2 + c) + (3*I*(2*a^3*b^2 - a*b^4)*d^2*x^4 - 3*I*(2*a^3*b^2 - a*b^4)*c^2 + (3*I*(2*a^4*b - a^2*b^3)*d^2
*x^4 - 3*I*(2*a^4*b - a^2*b^3)*c^2)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log(1/2*(2*b*cos(d*x^2 + c) - 2*I*
b*sin(d*x^2 + c) + 2*(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a) + 2*(6*(a^2*b^3
- b^5)*d*x^2 + 6*(a^2*b^3 - b^5)*c + 6*((a^3*b^2 - a*b^4)*d*x^2 + (a^3*b^2 - a*b^4)*c)*cos(d*x^2 + c) + (-3*I*
(2*a^3*b^2 - a*b^4)*d^2*x^4 + 3*I*(2*a^3*b^2 - a*b^4)*c^2 + (-3*I*(2*a^4*b - a^2*b^3)*d^2*x^4 + 3*I*(2*a^4*b -
a^2*b^3)*c^2)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))*log(1/2*(2*b*cos(d*x^2 + c) - 2*I*b*sin(d*x^2 + c) - 2*
(a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3*cos
(d*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (a + b \sec{\left (c + d x^{2} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*sec(d*x**2+c))**2,x)

[Out]

Integral(x**5/(a + b*sec(c + d*x**2))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^5/(b*sec(d*x^2 + c) + a)^2, x)